Analytical investigation on free torsional vibrations of noncircular nanorods
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(2020) 42:514
TECHNICAL PAPER
Analytical investigation on free torsional vibrations of noncircular nanorods Farshad Khosravi1 · Seyed Amirhosein Hosseini2 · Babak Alizadeh Hamidi3 Received: 16 November 2019 / Accepted: 25 August 2020 © The Brazilian Society of Mechanical Sciences and Engineering 2020
Abstract This paper is devoted to the free torsional behavior of the nanorods containing noncircular cross sections. The rectangular cross section is chosen to be the case of the study. Three various boundary conditions, namely the clamped–clamped (C–C), clamped–free (C–F), and clamped–torsional spring (C–T) boundary conditions, are used to model the nanorod. Hamilton’s principle is utilized to derive the equation of motion along with associated boundary conditions. The derived equation is reformulated by Eringen’s nonlocal elasticity approach to exhibit the small-scale effect. An analytical method is established to discretize and analyze the equation of motion. The novelty of this work is the analysis of the torsional vibration in rectangular nanorods, which are not observed in previous works. For the results, the influences of the horizontal and vertical aspect ratios ( a∕b and b∕a ) (for C–C and C–F boundary conditions) and the influences of the nonlocal parameter and stiffness of the boundary spring (for C–T boundary condition) are illustrated schematically and tabularly. Keywords Torsional vibration · Free vibration · Noncircular vibration · Rectangular cross section · Nanorod
1 Introduction Since the introduction of carbon nanotubes (CNTs) by Iijima [1, 2], one of the most important issues has been concerned with the vibration of the CNTs in the last two decades. Thus, many studies have been devoted to this field. The output of these investigations has led to benefit from these nanoscale structures in nanocomposites [3, 4], nanoreactors [5], nanoribbons [6, 7], nanoresonators [8, 9], nanosensors [10, 11], and nano-optics [12, 13]. Nano- and microstructures are divided into different categories, such as nano-/microrods, nano-/microbeams, nano-/microshells, nano-/microbars, and nano-/microplates [14]. Technical Editor: Thiago Ritto. * Seyed Amirhosein Hosseini [email protected] 1
Department of Aerospace Engineering, K.N. Toosi University of Technology, Tehran, Iran
2
Department of Industrial, Mechanical and Aerospace Engineering, Buein Zahra Technical University, Buein Zahra, Qazvin, Iran
3
Department of Mechanical Engineering, University of Tabriz, Tabriz, Iran
The nonlocal Eringen’s elasticity theory [15–19] has been established to show the small-scale effect and the torsional behavior of the nanorods. In this way, many studies are devoted to the vibration of the nanostructure via the mentioned nonlocal approach [20–31]. Meanwhile, there are some investigations that are carried out via the other theories such as Refs. [32–34]. Li and Hu [35] employed a closedform solution to investigate the free torsional behavior of bidirectional FG nanotubes based on Eringen’s nonlocal elasticity theory.
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